Think that a coin toss is 50-50 heads or tails? Read on to appreciate the ever-changing and random nature of the world in which we live.

Scale Invariance

Stage: 5 Challenge Level:

Why do this problem?

This problem offers a fascinating exploration into probability
density functions for real world data. Whilst the individual steps
are quite simple, the problem draws together many strands from
distribution theory. The results can be tested on any set of data
from any geography book, giving an interesting relevance to the
mathematics.

Possible approach

The first obstacle to overcome is that of notation: can the
students understand what is being asked?

The question involves little computation but requires clear
thinking of the ideas. This might be facilitated in a group
discussion, but might also require individual work.

Key questions

If a function is to be a probability density function, what is
the major property it must possess?

What ranges of values will start with a digit $1$?

Possible extension

Consider carefully why this problem involves 'scale invariance'.
Consider the restriction of scale invariance on real world data.
Which sets of real world data do you think will be modelled by this
distribution? Why?

Possible support

Skip the first part and provide students with the scale invariant
functions. Also, first use the range 1< x < 10 in the last
part of the question.

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the
NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to
embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities
can be found here.